I will sketch a construction which, among other things, relates homomorphisms of
certain abelian varieties over a field k to Mordell-Weil groups of certain abelian varieties over K=k(t). When k is finite this leads to elliptic curves of unbounded rank over k(t) with explicit generators for the Mordell-Weil group. When k is the complex numbers, it leads to (quite non-explicit) elliptic curves over C(t) of moderately high rank.